With his triumph on Monday, Federer, victor of the 2018 Australian Open (20th Grand Slam title) and the Rotterdam tournament, extends his winning streak this year to 14-0.
"It's been a long time since I have been interrupted at night and have to come back the next day", said Federer.
Federer returned to stadium court on Sunday afternoon and needed an hour to close out his second-round victory 6-3, 7-6 (8/6).
What he saw was his 36-year-old father overcome a shaky first set in which there were four breaks of serve. It should be noted that Federer is bidding to snap his own mark as the oldest player to ever win the tournament, after setting the record a year ago.
Federer served six aces and broke the Krajinovic serve five times.
"In Rotterdam I played five matches in a row", he said. "I felt I was too close to winning".
"I was grateful to be out on the court after surgery that quickly but at the same time just didn't feel good at all", he added.
His third-round opponent is Argentina's Leonardo Mayer, who had been scheduled to play Kei Nishikori in the second round before the world No. 25 withdrew through illness.
In the next round he faces Jeremy Chardy, who beat fellow Frenchman Adrian Mannarino 7-5, 4-6, 6-1.
Rain - a rarity in the desert - delayed the start of the match, with Federer cautious but restless to get play underway in front of a capacity crowd. Cuevas was leading 3-6, 6-4, 4-2 when Thiem packed it in.
Seventh seed Kevin Anderson took down Nicolas Kicker 7-6 (1) 7-6 (3) and South Korea's Chung Hyeon beat Tomas Berdych 6-4 6-4.
There were a few other upsets in Monday's action, including unseeded Croatian Borna Coric taking down the 13th seed, Spain's Roberto Bautista Agut, 6-1, 6-3 in 81 minutes.